Theorem rnun 5006

Description: Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)

Assertion

Ref	Expression
rnun	|- ran ( A u. B ) = ( ran A u. ran B )

Proof of Theorem rnun

Step	Hyp	Ref	Expression
1		cnvun 5003	. . . 4 |- `' ( A u. B ) = ( `' A u. `' B )
2	1	dmeqi 4799	. . 3 |- dom `' ( A u. B ) = dom ( `' A u. `' B )
3		dmun 4805	. . 3 |- dom ( `' A u. `' B ) = ( dom `' A u. dom `' B )
4	2, 3	eqtri 2185	. 2 |- dom `' ( A u. B ) = ( dom `' A u. dom `' B )
5		df-rn 4609	. 2 |- ran ( A u. B ) = dom `' ( A u. B )
6		df-rn 4609	. . 3 |- ran A = dom `' A
7		df-rn 4609	. . 3 |- ran B = dom `' B
8	6, 7	uneq12i 3269	. 2 |- ( ran A u. ran B ) = ( dom `' A u. dom `' B )
9	4, 5, 8	3eqtr4i 2195	1 |- ran ( A u. B ) = ( ran A u. ran B )

Syntax hints:   = wceq 1342   u. cun 3109   `'ccnv 4597   dom cdm 4598   ran crn 4599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-un 3115  df-in 3117  df-ss 3124  df-sn 3576  df-pr 3577  df-op 3579  df-br 3977  df-opab 4038  df-cnv 4606  df-dm 4608  df-rn 4609

This theorem is referenced by:  imaundi  5010  imaundir  5011  rnpropg  5077  fun  5354  foun  5445  fpr  5661  fprg  5662  sbthlemi6  6918  exmidfodomrlemim  7148